A characterization of a map whose inverse limit is an arc

TL;DR

This paper characterizes when the inverse limit of a continuous function on [0,1] forms an arc, using the concept of splitting sequences to provide a precise criterion.

Contribution

It introduces the notion of splitting sequences and establishes a necessary and sufficient condition for the inverse limit to be an arc.

Findings

Inverse limit of $f$ is an arc iff $f$ admits no splitting sequence.

Defines splitting sequences for continuous functions on [0,1].

Provides a complete characterization of inverse limits as arcs.

Abstract

For a continuous function $f : [0,1] \to [0,1]$ we define a splitting sequence admitted by $f$ and show that the inverse limit of $f$ is an arc if and only if $f$ does not admit a splitting sequence.